1Math 1b Practical — Eigenvalues and eigenvectors, IFebruar y 9, 2011Let A be a square matrix. A (right) eigenvector with corresponding eigenvalue λ is anonzero (column) vector e so thatAe = λe. (1)We say λ is an eigenvalue of A when (1) holds for at least one nonzero vector e.You will b e asked to find the eigenvalues and eigenvectors for various matrices . Thismeans find all eigenvalues , and for each eigenvalue λ, describe all corresponding eigenvec-tors.Note that λ is an eigenvalue of A if and only if A−λI is singular, s ince (1) is equivalentto (A −λI)e = 0. This in turn is equivalent to det(λI − A) = 0. The polynomialdet(λI − A)=(−1)ndet(A − λI)is caled the characteristic polynomial of A; the zeros of the charateristic polynomial arethe eigenvalues of A.Givenaneigenvalueλ, the subspaceUλ= {e : Ae = λe} = null space of A − λIis called the eigenspace of A corresponding to λ, and its dimension is called the geometricmultiplicity of λ as an eigenvalue of A. The nonzero elements of Uλare exactly the eigen-vectors with corresponding eigenvalue λ,and0. To describe the eigenvectors correspondingto λ, we may give a basis for the eigenspace Uλ.Example 1. LetA =1342 .Ascalarλ is an eigenvalue of A if and only if0=det(A − λI)=1 − λ 342−λ=(1− λ)(2 −λ) − 12 = λ2− 3λ − 10.This equation has two ro ots, λ1=5andλ2= −2.What is the eigenspace correp onding to the eigenvalue 5? It is the null space ofA − 5I =−434 −3 .The matrix has rank 1 so the null space (the set of vectors orthogonal to the rows) hasdimension 1. We may take e1=(3, 4) as a basis.2What is the eigenspace correp onding to the eigenvalue −2? It is the null space ofA +2I =3344 .The matrix has rank 1 so the null space (the set of vectors orthogonal to the rows) hasdimension 1. We may take e2=(1, −1) as a b as is.Let us check our work:Ae1=1342 34 =1520 =5e1,Ae2=1342 1−1 =−22 = −2e2.Example 2. LetA =⎛⎜⎜⎜⎝6000006000006000007000007⎞⎟⎟⎟⎠.The characteristic polynomial of A is (λ − 6)3(λ − 7)2, so the eigenvalues are 6 and 7.What is the eigenspace correp onding to the eigenvalue 6? It is the null space ofA − 6I =⎛⎜⎜⎜⎝0000000000000000001000001⎞⎟⎟⎟⎠.The matrix has rank 2 so the null space of A − 6I has dimension 3. We may take(1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0) as a basis.What is the eigenspace correp onding to the eigenvalue 7? It is the null space ofA − 7I =⎛⎜⎜⎜⎝−10 0000 −100000−1000000000000⎞⎟⎟⎟⎠.The matrix has rank 3 so the null space of A − 7I has dimension 2. We may take(0, 0, 0, 1, 0), (0, 0, 0, 0, 1) as a bas is.Example 3. LetA =⎛⎜⎜⎜⎝6100006100006000007100007⎞⎟⎟⎟⎠.3The characteristic polynomial of A is (λ − 6)3(λ − 7)2, so the eigenvalues are 6 and 7.What is the eigenspace correp onding to the eigenvalue 6? It is the null space ofA − 6I =⎛⎜⎜⎜⎝0100000100000000001100001⎞⎟⎟⎟⎠.The m atrix has rank 4 so the null space of A−6I has dimens ion 1. We may take (1, 0, 0, 0, 0)as a b as is.What is the eigenspace correp onding to the eigenvalue 7? It is the null space ofA − 7I =⎛⎜⎜⎜⎝−11 0000 −110000−1000000100000⎞⎟⎟⎟⎠.This matrix has rank 4 so the null space of A−7I has dimension 1. We may take (0, 0, 0, 1, 0)as a b as is.Example 4. LetA =⎛⎜⎝1234234534564567⎞⎟⎠.The characteristic polynomial of A is x4− 16x3− 20x2(I found this with Mathematica),so the eigenvalues are 0 and the zer os of x2− 16x −20, which are 8 ± 2√21).What is the eigenspace correponding to the eigenvalue 0? It is the null space of Aitself. The reduced echelon form of A is⎛⎜⎝10−1 −201 2 300 0 000 0 0⎞⎟⎠.So a basis for the null space of A is (1, −2, 1, 0), (2, −3, 0, 1). The eigenvectors of A corre-sponding to the other eigenvalues are a little messy. I found them with Mathematica. TheMathematica function Eigensystem[A] returns a list of the eigenvalues of A followed bya list of corresponding eigenvectors. To see the results in decimal approximations, I typed1.0 %, which multiplies the immediately preceding output by 1.0; the decimal point forcesthe display in floating point.4Example 5. LetA =⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝3 −38−97 3 6−33−37 −493445−15−1−16−7 −8 −2 − 5 −23−4 −660−6 −16 7 6−4 −65−58 9−1 −39−7 −53 734−41 4−8 − 2 −4 −6 −20563−69−2 −82−574−1 −5 −29345−6−96 0−2 −5 −1−1 −1 −2134200−59−55−4⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.There are 10 distinct eigenvalues, the floating point approximations of which are{-17.8334,-12.1945+4.97853 I,-12.1945-4.97853 I,-1.28814+10.6727 I,-1.28814-10.6727 I,10.5175,4.07423+9.37028 I,4.07423-9.37028 I,1.06641+3.30617 I, 1.06641-3.30617 I}.Note that many of these eigenvalues are complex numbers. This is an instance wherecomplex numbers are required even if we are talking ab out real matrices. The correspond-ing eigenvectors will also be